# Image Geometry Requirements

Name:
Image Geometry Requirements
Date:
12 March 2008

## The image cosines are basis vectors

The direction cosine data elements in Image Orientation (Patient) (0020,0037) represent two of the three basis vectors that define the directions of the image slice axes. The first three values are the components of the image slice row vector

$r = \left[r_x r_y r_z \right]$

The next three are the components of the image slice column vector

$c = \left[c_x c_y c_z \right]$

The third basis vector of the image slice, the normal, is the cross product of the row and column vector

$n = \left[n_x n_y n_z \right] = r \times c$

The meaning of the image requirements The direction cosines shall be normalized. The image axis basis vectors shall be unit vectors to within δ, on the order of the precision of floating-point arithmetic

$\begin{matrix} \left | \left \| r \right \| - 1 \right | < δ \\ \left | \left \| c \right \| - 1 \right | < δ \\ \left | \left \| n \right \| - 1 \right | < δ \end{matrix}$

The image axes shall be orthogonal

$\begin{matrix} r \cdot c = 0 \\ r \cdot n = 0 \\ r \cdot n = 0 \end{matrix}$ 

The image axes shall be parallel or anti-parallel with respect to the axes of the patient coordinate system. Given the unit basis vectors of the axes of the patient coordinate system

$i = \left[ \: 1 \: 0 \: 0 \: \right] \\ j = \left[ \: 0 \: 1 \: 0 \: \right] \\ k = \left[ \: 0 \: 0 \: 1 \: \right]$

and a small deviation, ε, we require

$\left| r \cdot i \right] > 1 - ε$ and $\left| c \cdot j \right] > 1 - ε$

for prone and supine patient orientations,

$\left| r \cdot j \right] > 1 - ε$ and $\left| c \cdot i \right] > 1 - ε$

for decubitus patient orientations, and

$\left| n \cdot k \right] > 1 - ε$

in either case.